1 NORMED ALGEBRAS 309 3. Let ^(X) be the set of continuous functions on the closed unit disk X : | z\ <^ 1 in C which are analytic in the interior |z| < 1 of X. For each pair of elements x, y e we define for all £ e X. Show that this product makes J/(X) a commutative algebra without a unit element, and that with respect to the norm induced by the norm on ^C(X) the algebra Ğ^(X) is a Banach algebra. 4. (a) Let O be the set of finite real-valued functions CD on R belonging to LJ(R,A) (where A is Lebesgue measure) such that o>( 0= &>(0, w(0^0 for all reR, and such that oğ is decreasing on [0, + oo [. Every function a> e Q, is the uniform limit of an increasing sequence (o>n) of step functions belonging to 0. Put V(cu) = co(0) + f o>(0 dt. J ~ oo If o>i, o>2 e £1, show that co* * o>2 e Q, and that V(ct>! * o>2) ^ V(o>i) V(o>2) (consider first the case where one of wlt co2 is a step function). (b) Let $2 denote the set of A-measurabie complex-valued functions / on R such that, for at least one function co e Q, the function \f\2(^~l is A-integrable (with the conven- tion that 0 (+ oo) = 0). We have Q c j/. For each function/e *af, put f+0° I/WI2 V/2 infV(o)) J^7;r^ 6)6n J-oo w(/) / r + °° \f(t)\2 \ ii77r^ J-oo cu(/) / Show that also NB(/) = ^n Deduce that <$# is a complex vector space and that NB is a seminorm on A. (Observe that if a, b, a, /? are real and >0, then Show that j* <= ^(R, A) n ^(R, A) and that NI(/) g NB(/) and N2(/) ^NB(/). (c) Let/e ^f1 n ^f2 be the function defined as follows : f(x) = 1/n2 if 2n - 1 ^ x ^ 2" (w an integer ^l),/(x) = 0 otherwise. Show that/^ ^/. (d) For each function /e ^, show that there exists a>f e Q such that (Consider a sequence (o>n) of functions belonging to O such that V(ei>n) = NB(/) and such that the sequence of integrals |/|2o>n~l dX tends to NB(/). Then consider the function lim sup wn , and use Fatou's lemma.) n-^oo (e) The subspace Jf of A-negligible functions is contained in ^. By passing to the quotient, NB induces a norm on the space A = jtf/*^. Show that, with respect to this norm (denoted by NB(/) or ||/||), the space A is a Banach space. (If (/) is a sequence in oo